Closed Form Solutions of the Second Order Linear ODEs with Non-Constant Coefficients

Question

I am studying about the linear odes with non-constant coefficients.

I know the first order linear ode with non-constant coefficient

$$y^{'}(x)+f(x)y(x)=0 \tag{1}$$

has a general solution of the form

$$y=Ce^{-\int f(x) dx} \tag{2}$$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$$y^{''}(x)+g(x)y^{'}(x)+f(x)y(x)=0 \tag{3}$$

I know that this equation does not have a closed form solution like $(2)$. However, I am interested in special cases of that.

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Questions

$1$. Consider $(3)$, when $g(x)=0$, then we have

$$y^{''}(x)+f(x)y(x)=0 \tag{4}$$

Is Eq.$(4)$ a famous well-known equation? If YES, what is its name?

$2$. Does $(4)$ have a closed form solution like $(2)$?

$3$. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

Answer 1

Somewhere (but where ?) I already answer to a question quite the same as your question 1 (only change a sign in equation 4). By luck, I didn't remove my draft (copy below).

There is no general formula for the solutions of equation 4 in cases of any $f(x)$ since specific special functions are defined according to each specific case. All the more so for equation 3.

COPY :

$$y''(x)=f(x)y(x)$$

Case: $f(x)=c^2 \quad\to\quad y(x)=c_1e^{cx}+c_2e^{-cx}=c_3\cosh(cx)+c_4\sinh(cx)$

Case: $f(x)=-c^2 \quad\to\quad y(x)=c_1\cos(cx)+c_2\sin(cx)$

Case: $f(x)=x \quad\to\quad y(x)=c_1Ai(x)+c_2Bi(x) \quad $ Ary functions.

Case: $f(x)=x^2 \quad\to\quad y(x)=c_1 D_{-1/2}(\sqrt{2}x) +c_2 D_{-1/2}(i\sqrt{2}x) \quad $ Parabolic cylinder function.

Case: $f(x)=-\lambda^2 x^{\frac{1}{\nu}-2} \quad\to\quad y(x)=c_1 \sqrt{x}J_{\nu}(\lambda x) +c_2 \sqrt{x}Y_{\nu}(\lambda x) \quad $ Bessel functions.

Case: $f(x)=\lambda^2 x^{\frac{1}{\nu}-2} \quad\to\quad y(x)=c_1 \sqrt{x}I_{\nu}(\lambda x) +c_2 \sqrt{x}K_{\nu}(\lambda x) \quad $ Modified Bessel functions.

Case: $f(x)=-a+2b\cos(2x) \quad\to\quad y(x)= c_1C(a\:,\:b\:;\:x)+c_1S(a\:,\:b\:;\:x)$ Mathieu functions.

Case: $f(x)=\frac{A}{x^2}+\frac{B}{x}+C \quad\to\quad y(x)=e^{-\frac{\gamma}{2}x}x^{\frac{\beta}{2}}\left( c_1 M(\alpha\:,\:\beta\:;\:\gamma x)+c_2 U(\alpha\:,\:\beta\:;\:\gamma x) \right) \quad$ with $\begin{cases} \gamma=\pm2\sqrt{C}\\ \beta=1\pm 2\sqrt{A+\frac{1}{4}}\\ \alpha=\frac{\beta}{2}+\frac{B}{\gamma} \end{cases}\quad$ Kummer and Tricomi functions (confluent hypergeometric functions).

Etc.